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76
A NEW TYPE MAGIC LATIN 3~CUBE OF ORDER TEN
[Feb.
give a more interesting example, consider the geometric progression {ak} which
constitutes a Benford sequence in base b if and only if a ^ bvlq (p, q integers).
Setting a - 3 and br = 9, we obtain a subset of the positive integers which is
not a Benford sequence. Moreover, Pr(j < 4) 9 = 1 for the geometric progression
{3^}. Since {3k} is a Benford sequence in base b = 8, we may apply Lemma 1 with
a = 2,77Z = 2, n = 3 t o yield Pr{j < 4 ) 8 = 2/3. A comparison of the above probabilities for b = 8 and bf = 9 shows that the monotonicity statement is false
for this example.
REFERENCES
1.
2.
3.
F. Benford. "The Law of Anomalous Numbers." Proc. Amer. Phil. Soc. 78 (1938):
551-557.
J.L, Brown & R. L. Duncan. "Modulo One Uniform Distribution of the Sequence
of Logarithms of Certain Recursive Sequences." The Fibonacci
Quarterly
8
(1970):482-486.
R.A. Raimi. "The First Digit Problem." Amer. Math. Monthly 83 (1976):521538.
A NEW TYPE MAGIC LATIN 3-CUBE OF ORDER TEN
JOSEPH ARKSN
197 Old Nyack Turnpike,
Spring
Valley,
NY 10977
K. SINGH
University
of New Brunswick,
Fredericton,
N*B. E3B 5A3,
Canada
A Latin 3-cube of order n is an n x n x n cube (n rows, n columns, and n
files) in which the numbers 0 9 1, 2, ..., n - 1 are entered so that each number
occurs exactly once in each row, column, and file. A magic Latin 3-cube of order n is an arrangement of n 3 integers in three orthogonal Latin 3-cubes, each
of order n (where every ordered triple 000, 001, ..., n - 1 , n - 1 , n - 1 occurs)
such that the sum of the entries in every row, every column, and every file, in
each of the four major diagonals (diameters) and in each of the n1 broken major
diagonals is the same; namely, hn(n3 + 1 ) . We shall list the cubes in terms of
n squares of order n that form its different levels from the top square 0 down
through (inclusively) square 1, square 2, ..., square n - 1 . We define a broken major diagonal as a path (route) which begins in square 0 and goes through
the n different levels (square 0, square 1, ..., square n - 1) of the cube and
passes through precisely one cell in each of the n squares in such a way that
no two cells the broken major diagonal traverses are ever in the same file.
The sum of the entries in the n cells that make up a broken major diagonal
equals hn(n3 + 1 ) . A complete system consists of n2 broken major diagonals,
where each broken major diagonal emanates from a cell in square 0, and thus the
n2 broken major diagonals traverse each of the n 3 cells of the cube in n2 distinct routes. The cube is initially constructed as a Latin 3-cube in which the
numbers are expressed in the scale of n (0, 1, 2, ..., n - 1). However, after
adding 1 throughout and converting the numbers to base 10, we have the n 3 numbers 1, 2, . .., ns where the sum of the entries in every row, every column,
and every file in each of the four major diagonals, and in each of the n2 broken major diagonals is the same; namely, hn(n3 + 1).
In this paper, for the first time in mathematics, we construct a magic Latin 3-cube of order ten. In this case, the sum of the numbers in every row,
1981]
A NEW TYPE MAGIC LATIN 3™CUBE OF ORDER TEN
77
every column, and every file in each of the four major diagonals, and in each
of the 102 broken major diagonals is the same; namely, %(10)(103 + 1) = 5QG5«
In Chart 1 we list (by columns) the coordinates of the cells through which
10 broken major diagonals pass. It should be noted that the first digit of the
coordinates denotes the row3 the second digit the column, and at the right side
of each row is the square number in which each cell is to be found. Each one of
the 10 broken major diagonals is found under one of the 10 columns, that is, In
Chart 1 we find listed by columns 10 broken major diagonals, where each column
denotes one broken major diagonal,, For example, under column 0, we find the
coordinates 00, 99, 55, 66, 11, 88, 77, 22, 44, and 33. These cells determine
one broken major diagonal. After adding 1 to each number found in the corresponding 10 cells in the 10 squares, we get
764+373+791 + 588+707+026+445+340+612+359 - ^(10) (103 + 1) = 5005.
Now, in order to find the remaining 90 broken major diagonals that emanate
from square 0, we must construct nine more charts to get Chart 1, Chart 2, . .« ,
Chart J. We need only show (as an example) how to construct Chart 2 from Chart
1 and the Key Chart, since the remaining eight charts (Chart 2, ..., Chart X)
are constructed in exactly the same way,
In the Key Chart under column I are the numbers In the same order that are
found in Chart 1 under column 0«
In Chart 1, we define the rows as follows:
a (00) row = 00
a (99) row = 99
16
41
29
62
35
56
42
84
53
75
64
23
71
10
87
08
98
37
square 0
square 1
a (33) row - 33
28
45
04
79
86
90
57
61
12
square 9
Thus, in the Key Chart we have under column I a (0.0) row, a (99) row,...*, and
a (33) row, which is, of course, a restatement in a shorter form of the entire
Chart 1.
Now, in the Key Chart, each number under column II which is identical to a
number under column I (in the Key Chart) represents the identical row found In
Chart 1. Therefore, Chart 2 is written as:
(11) row:
(33) row:
11
33
89
28
76
45
27
04
58
79
94
86
32
90
03
57
40
61
65
12
square 0
square 1
(99) row:
99
41
62
56
84
75
23
10
08
37
square 9
Then the columns of Chart 2 give 10 more broken major diagonals,
We can find the remaining eight charts—Chart 3, Chart 4, . .., Chart J—in
exactly the same way as Chart 2, using the Key Chart In conjunction with Chart
1. (The charts are presented on the following pages.)
It should be noted here that Chart 1 is constructed by superposing two orthogonal Latin squares of order ten. Now, since it is impossible to superpose
two Latin squares of order n when n = 2 or 6, we may state that this type of
magic Latin 3-cube is impossible for order 2 and for order 6.
In the near future, we shall present a more comprehensive general paper In
which we consider the general order km + 2 and the powers of prime numbers.
78
A NEW TYPE MAGIC LATIN 3-CUBE OF ORDER TEN
[Feb.
CEAB.T 1
0
1
00
,99
55
66
11
88
77
22
44
33
16
29
41
62
97
38
50
93
89
76
72
01
34
80
05 " 54
63
17
28
45
2
3
35
56
19
82
27
43
68
70
91
04 "
4
42
84
60
07"
58
95
13
31
26
79
5
6
53
75
02
21
94
67
49
18
30
86
KEY CHART FOR 100
7
64
23
81
15
32
59
06
47
78
90
71
10
24
48
03
36
92
69
85
57
8
9
87
98
square 0
08
37
square 1
73 . 46 = square 2
39
74
square 3
40
65
square 4
14
20
square 5
25
51
square 6
96
83
square 7
52
09
square 8
61
12
square 9
BROKEN MAJOR DIAGONALS
I
II
III
IV
V
VI
VII VIII
IX
X
00
11
22
33
44
55
66
77
88
99
square 0
22
00
77
55
11
66
square 1
88
99
66
44
square 2
square 3
99
33
88
44
55
22
33
00
77
11
66
44
77
55
00
88
99
1.1
33
22
11
00
99
77
66
44
33
22
55
88
square 4
88
55
11
66
99
33
22
44
00
77
square 5
77
88
66
99
11
22
55
00
44
33
square 6
22
55
99
00
33
77
11
square 7
66
44
88
44
77
00
22
33
66
11
88
99
55
square 8
33
99
55
11
88
77
44
66
22
00
square 9
MAGIC LATIN
Square
0
1
2
3
4
5
6
7
8
9
3-CUBE OF ORDER TEN
Number 0
0
1
2
3
4
5
6
7
8
9
763
279
897
140
932
328
554
415
686
001
886
963
340
454
228
697
032
779
501
115
540
097
463
901
754
132
286
828
315
679
979
654
201
063
815
740
128
532
497
386
015
832
579
628
163
486
701
397
254
940
428
301
632
715
086
563
997
240
179
854
601
728
154
879
597
354
186
915
297
401
079
840
663
732
528
232
440
028
586
379
954
615
101
863
797
197
515
786
332
640
801
479
054
928
263
21.5
363
986
040
432
1981]
A NEW TYPE MAGIC LATSN 3-CUBE OF ORDER TEN
Square
0
1
2
3
4
5
6
7
8
9
Number 1
0
1
2
3
4
5
6
7
8
9
472
385
100
564
059
816
221
693
938
747
138
072
864
621
316
900
759
485
247
593
264
700
672
047
421
559
338
116
893
985
085
921
347
772
193
464
516
259
600
838
793
159
285
916
572
638
447
800
321
064
616
847
959
493
738
272
000
364
585
121
947
416
521
185
200
393
872
038
764
659
821
538
093
300
647
785
164
972
459
216
359
664
716
238
885
021
993
547
172
400
500
293
438
859
964
147
685
721
016
372
Square
0
1
2
3
4
5
6
7
8
9
Number 2
0
1
2
3
4
5
6
7
8
9
190
413
952
071
346
665
737
508
224
889
924
390
671
537
465
252
846
113
789
008
771
852
590
389
137
046
424
965
608
213
313
237
489
890
908
171
065
746
552
624
808
946
713
265
090
524
189
652
437
371
565
689
246
108
824
790
352
471
013
937
289
165
037
913
752
408
690
324
781
546
637
024
308
452
589
813
971
290
146
765
446
571
865
724
613
337
208
089
990
152
052
708
124
646
271
989
513
837
365
490
Square
0
1
2
3
4
5
6
7
8
9
79
Number 3
0
1
2
3
4
5
6
7
8
9
987
131
266
323
418
502
875
049
750
694
250
487
523
075
102
766
618
931
894
349
823
666
087
494
975
318
150
202
549
731
431
775
194
687
249
923
302
818
066
550
649
218
831
702
387
050
994
566
175
423
002
594
718
949
650
887
466
123
331
275
794
902
375
231
866
149
587
450
623
018
575
350
449
166
094
631
223
787
918
802
118
023
602
850
531
475
749
394
287
966
366
849
950
518
723
294
031
675
402
187
80
A NEW TYPE MAGSC LATIN 3~CUBE OF ORDER TEN
Square
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
606
827
578
255
783
199
362
934
041
410
541
706
115
962
899
078
483
627
310
234
355
478
906
710
662
283
841
599
134
027
727
062
810
406
534
655
299
383
978
141
434
583
327
099
206
999
110
083
634
441
306
778
855
227
562
010
699
262
527
378
834
106
741
455
983
162
241
734
878
910
427
555
006
683
399
883
955
499
341
127
762
034
210
506
678
278
334
641
183
055
510
927
462
799
806
0
1.
2
3
4
5
6
7
8
9
525
642
014
788
891
930
403
257
369
176
069
825
988
203
630
314
191
542
476
757
488
114
225
503
791
669
030
957
342
842
303
676
125
057
588
730
491
214
969
157
091
442
330
725
269
576
914
603
888
230
976
391
557
169
425
814
688
742
003
376
530
703
042
414
657
925
869
188
291
903
769
857
614
276
142
088
325
591
430
691
288
130
469
942
803
357
776
025
514
714
457
569
991
388
076
242
103
830
625
0
1
2
3
4
5
6
7
8
9
044
598
329
836
605
277
180
761
412
953
312
644
236
780
577
429
905
098
153
861
136
929
744
653
080
805
512
377
261
498
698
480
553
944
361
036
877
105
729
212
961
305
198
477
844
712
053
229
580
636
111
253
405
061
912
144
629
536
989
380
453
077
880
398
129
561
244
612
936
705
280
812
661
529
753
998
336
444
005
177
505
736
977
112
298
680
461
853
344
029
829
161
012
205
436
353
798
980
677
544
94.1
610
178
862
755
786-
Square
0
•1
2
3
4
5
6
7
8
9
Number 4
0
Square
0
1
2
3
4
5
6
7
8
9
[Feb
Number 5
Number 6
1981]
A NEW TYPE MAGIC LATIN 3~CUBE OF ORDER TEN
Square
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
239
904
745
417
120
051
696
382
873
568
773
139
017
396
951
845
520
204
668
482
617
545
339
168
296
420
973
751
082
804
104
986
968
539
782
217
451
620
345
073
582
720
604
851
439
373
268
045
996
117
351
068
820
282
573
639
145
917
404
796
868
251
496
704
645
982
039
173
517
320
096
473
182
945
368
504
717
839
220
651
920
317
551
673
004
196
882
468
739
245
445
682
273
020
817
768
304
596
151
939
6
7
8
9
122
384
648
456
933
070
711
595
209
867
748
695
570
033
822
256
409
111
367
984
067
809
284
995
756
548
170
622
411
333
633
970
395
767
109
422
856
248
584
1
2
3
4
5
6
7
8
9
Number 8
0
1
2
3
4
311
056
433
609
567
784
948
870
195
222
495
511
709
848
084
133
267
356
922
670
909
233
811
522
348
667
095
484
770
156
556
148
022
211
470
309
684
967
833
795
270
467
956
184
611
895
322
733
048
509
0
1
2
858
760
681
992
274
443
019
126
507
335
607
258
492
119
743
581
374
860
035
926
092
381
158
235
819
974
707
643
426
560
Square
0
Number 7
0
Square
0
81
5 '
884
722
167
370
295
911
533
009
656
448
on
Number 9
3
4
5
6
7
8
9
260
519
735
358
626
892
943
074
181
407
326
674
060
543
958
107
835
481
719
292
143
435
574
826
307
058
281
792
960
619
535
843
919
660
081
726
458
207
392
174
419
907
226
781
135
360
692
558
874
043
774
192
343
007
460
219
526
935
658
881
981
026
807
474
592
635
160
319
243
758
82
COMPLEX FIBONACCI NUMBERS
[Feb.
REFERENCES
Joseph Arkin & E. G. Straus. "Latin /c-Cubes." The Fibonacci
Quarterly
12
(1974)^288-292.
J. Arkin. "A Solution to the Classical Problem of Finding Systems of Three
Mutually Orthogonal Numbers in a Cube Formed by Three Superimposed 10 x 10
x 10 Cubes." The Fibonacci
Quarterly
12 (1974):133-140. Also, Sugaku Seminar 13 (1974):90-94.
R. C. Bose, S. S. Shrikhande, & E. T. Parker. "Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of EulerTs
Conjecture." Canadian J'. Math. 12 (1960) : 189-203.
COMPLEX FIBONACCI NUMBERS
Mitchell
College
C. J. HARMAN
Education,
Bathurst
of Advanced
1.
N.S.W.
2795,
Australia
INTRODUCTION
In this note9 a new approach is taken toward the significant extension of
Fibonacci numbers into the complex plane. Two differing methods for defining
such numbers have been considered previously by Horadam [4] and Berzsenyi [2].
It will be seen that the new numbers include Horadam1s as a special case, and
that they have a symmetry condition which is not satisfied by the numbers considered by Berzsenyi.
The latter defined a set of complex numbers at the Gaussian integers, such
that the characteristic Fibonacci recurrence relation is satisfied at any horizontal triple of adjacent points. The numbers to be defined here will have the
symmetric condition that the Fibonacci recurrence occurs on any horizontal or
vertical triple of adjacent points.
Certain recurrence equations satisfied by the new numbers are outlined, and
using them, some interesting new Fibonacci identities are readily obtained.
Finally, it is shown that the numbers generalize in a natural manner to higher
dimensions.
2.
THE COMPLEX FIBONACCI NUMBERS
The numbers, to be denoted by G(n9 m), will be defined at the set of Gaussian integers (ji9 m) = n + im9 where n e TL and m e 7L . By direct analogy with
the classical Fibonacci recurrence
F
«+2
=Fn+l
+ P
n , F0
= 0,
F,
= 1,
(2.1)
the numbers G(n9 rri) will be required to satisfy the following two-dimensional
recurrence
Gin
G(n9
where
+ 2, rri) = Gin
+ 1, rri) + G(n9
rri),
m + 2) = (?(n, m + 1) + G(n9 rri) 9
£(0, 0) = 0, G(l9
0) = 1, G(Q9 1) = i9
G(l9
1) = 1 + '£.
(2.2)
(2.3)
(2.4)
The conditions (2.2), (2.3), and (2.4) are sufficient to specify the unique
value of Gin9 rri) at each point (n, rri) in the plane, and the actual value of
G(n9 rri) will now be obtained.
From (2.2), the case m = 0 gives
Gin + 2, 0) - Gin + 1, 0) + G(n9 0 ) ; Gi09
0) = 0, G(l, 0) = 1